Download The Capital Asset Pricing Model

ECON 337901
FINANCIAL ECONOMICS
Peter Ireland
Boston College
Spring 2017
These lecture notes by Peter Ireland are licensed under a Creative Commons Attribution-NonCommerical-ShareAlike
4.0 International (CC BY-NC-SA 4.0) License. http://creativecommons.org/licenses/by-nc-sa/4.0/.
Two Perspectives on Asset Pricing
Where we have been . . .
A Introduction
1 Mathematical and Economic Foundations
2 Overview of Asset Pricing Theory
B Decision-Making Under Uncertainty
3 Making Choices in Risky Situations
4 Measuring Risk and Risk Aversion
C The Demand for Financial Assets
5 Risk Aversion and Investment Decisions
6 Modern Portfolio Theory
Two Perspectives on Asset Pricing
. . . and where we are heading:
D Classic Asset Pricing Models
7 The Capital Asset Pricing Model
8 Arbitrage Pricing Theory
E Arrow-Debreu Pricing
9 Arrow-Debreu Pricing: Equilibrium
10 Arrow-Debreu Pricing: No-Arbitrage
F Extensions
11 Martingale Pricing
12 The Consumption Capital Asset Pricing Model
7 The Capital Asset Pricing Model
A
B
C
D
MPT and the CAPM
Deriving the CAPM
Valuing Risky Cash Flows
Strengths and Shortcomings of the CAPM
MPT and the CAPM
The Capital Asset Pricing Model builds directly on Modern
Portfolio Theory.
It was developed in the mid-1960s by William Sharpe (US,
b.1934, Nobel Prize 1990), John Lintner (US, 1916-1983), and
Jan Mossin (Norway, 1936-1987).
MPT and the CAPM
William Sharpe, “Capital Asset Prices: A Theory of Market
Equilibrium Under Conditions of Risk,” Journal of Finance
Vol.19 (September 1964): pp.425-442.
John Lintner, “The Valuation of Risk Assets and the Selection
of Risky Investments in Stock Portfolios and Capital Budgets,”
Review of Economics and Statistics Vol.47 (February 1965):
pp.13-37.
Jan Mossin, “Equilibrium in a Capital Asset Market,”
Econometrica Vol.34 (October 1966): pp.768-783.
MPT and the CAPM
But whereas Modern Portfolio Theory is a theory describing
the demand for financial assets, the Capital Asset Pricing
Model is a theory describing equilibrium in financial markets.
By making an additional assumption – namely, that supply
equals demand in financial markets – the CAPM yields
additional implications about the pricing of financial assets and
risky cash flows.
MPT and the CAPM
Like MPT, the CAPM assumes that investors have
mean-variance utility and hence that either investors have
quadratic Bernoulli utility functions or that the random returns
on risky assets are normally distributed.
Thus, some of the same caveats that apply to MPT also apply
to the CAPM.
For example, one might hesitate before applying the CAPM to
price options.
MPT and the CAPM
The traditional CAPM also assumes that there is a risk free
asset as well as a potentially large collection of risky assets.
Under these circumstances, as we’ve seen, all investors will
hold some combination of the riskless asset and the tangency
portfolio: the efficient portfolio of risky assets with the highest
Sharpe ratio.
MPT and the CAPM
But the CAPM goes further than the MPT by imposing an
equilibrium condition.
Because there is no demand for risky financial assets except to
the extent that they comprise the tangency portfolio, and
because, in equilibrium, the supply of financial assets must
equal demand, the market portfolio consisting of all existing
financial assets must coincide with the tangency portfolio.
In equilibrium, that is, “everyone” must “own the market.”
Deriving the CAPM
In the CAPM, equilibrium in financial markets requires the
demand for risky assets – the tangency portfolio – to coincide
with the supply of financial assets – the market portfolio.
Deriving the CAPM
The CAPM’s first implication is immediate: the market
portfolio is efficient.
Deriving the CAPM
The line originating at (0, rf ) and running through
(σM , E (˜
rM )) is called the capital market line (CML).
Deriving the CAPM
Hence, it also follows that all individually optimal portfolios
are located along the CML and are formed as combinations of
the risk free asset and the market portfolio.
Deriving the CAPM
Recall that the trade-off between the standard deviation and
expected return of any portfolio combining the riskless asset
and the tangency portfolio is described by the linear
relationship
E (˜
rT ) − rf
E (˜
rP ) = rf +
σP .
σT
Since the CAPM implies that the tangency and market
portfolios coincide, the formula for the Capital Market Line is
likewise
E (˜
rM ) − rf
E (˜
rP ) = rf +
σP .
σM
Deriving the CAPM
And since all individually optimal portfolios are located along
the CML, the equation
E (˜
rM ) − rf
E (˜
rP ) = rf +
σP .
σM
implies that the market portfolio’s Sharpe ratio
E (˜
rM ) − rf
σM
measures the equilibrium price of risk: the expected return
that each investor gives up when he or she adjusts his or her
total portfolio to reduce risk.
Deriving the CAPM
Next, let’s consider an arbitrary asset – “asset j” – with
random return r˜j , expected return E (˜
rj ), and standard
deviation σj .
MPT would take E (˜
rj ) and σj as “data” – that is, as given.
The CAPM again goes further and asks: if asset j is to be
demanded by investors with mean-variance utility, what
restrictions must E (˜
rj ) and σj satisfy?
Deriving the CAPM
To answer this question, consider an investor who takes the
portion of his or her initial wealth that he or she allocates to
risky assets and divides it further: using the fraction w to
purchase asset j and the remaining fraction 1 − w to buy the
market portfolio.
Note that since the market portfolio already includes some of
asset j, choosing w > 0 really means that the investor
“overweights” asset j in his or her own portfolio. Conversely,
choosing w < 0 means that the investor “underweights” asset
j in his or her own portfolio.
Deriving the CAPM
Based on our previous analysis, we know that this investor’s
portfolio of risky assets now has random return
r˜P = w r˜j + (1 − w )˜
rM ,
expected return
E (˜
rP ) = wE (˜
rj ) + (1 − w )E (˜
rM ),
and variance
2
σP2 = w 2 σj2 + (1 − w )2 σM
+ 2w (1 − w )σjM ,
where σjM is the covariance between r˜j and r˜M .
Deriving the CAPM
E (˜
rP ) = wE (˜
rj ) + (1 − w )E (˜
rM ),
2
+ 2w (1 − w )σjM ,
σP2 = w 2 σj2 + (1 − w )2 σM
We can use these formulas to trace out how σP and E (˜
rP )
vary as w changes.
Deriving the CAPM
The red curve these traces out how σP and E (˜
rP ) vary as w
changes, that is, as asset j gets underweighted or
overweighted relative to the market portfolio.
Deriving the CAPM
The red curve passes through M, since when w = 0 the new
portfolio coincides with the market portfolio.
Deriving the CAPM
For all other values of w , however, the red curve must lie
below the CML.
Deriving the CAPM
Otherwise, a portfolio along the CML would be dominated in
mean-variance by the new portfolio. Financial markets would
no longer be in equilibrium, since some investors would no
longer be willing to hold the market portfolio.
Deriving the CAPM
Together, these observations imply that the red curve must be
tangent to the CML at M.
Deriving the CAPM
Tangent means equal in slope.
We already know that the slope of the Capital Market Line is
E (˜
rM ) − rf
σM
But what is the slope of the red curve?
Deriving the CAPM
Let f (σP ) be the function defined by E (˜
rP ) = f (σP ) and
therefore describing the red curve.
Deriving the CAPM
Next, define the functions g (w ) and h(w ) by
g (w ) = wE (˜
rj ) + (1 − w )E (˜
rM ),
2
h(w ) = [w 2 σj2 + (1 − w )2 σM
+ 2w (1 − w )σjM ]1/2 ,
so that
E (˜
rP ) = g (w )
and
σP = h(w ).
Deriving the CAPM
Substitute
E (˜
rP ) = g (w )
and
σP = h(w ).
into
E (˜
rP ) = f (σP )
to obtain
g (w ) = f (h(w ))
and use the chain rule to compute
g 0 (w ) = f 0 (h(w ))h0 (w ) = f 0 (σP )h0 (w )
Deriving the CAPM
Let f (σP ) be the function defined by E (˜
rP ) = f (σP ) and
therefore describing the red curve. Then f 0 (σP ) is the slope of
the curve.
Deriving the CAPM
Hence, to compute f 0 (σP ), we can rearrange
g 0 (w ) = f 0 (σP )h0 (w )
to obtain
f 0 (σP ) =
g 0 (w )
h0 (w )
and compute g 0 (w ) and h0 (w ) from the formulas we know.
Deriving the CAPM
g (w ) = wE (˜
rj ) + (1 − w )E (˜
rM ),
implies
g 0 (w ) = E (˜
rj ) − E (˜
rM )
Deriving the CAPM
2
+ 2w (1 − w )σjM ]1/2 ,
h(w ) = [w 2 σj2 + (1 − w )2 σM
implies
1
h0 (w ) =
2
(
2
2w σj2 − 2(1 − w )σM
+ 2(1 − 2w )σjM
2
2
2
2
[w σj + (1 − w ) σM + 2w (1 − w )σjM ]1/2
or, a bit more simply,
h0 (w ) =
2
w σj2 − (1 − w )σM
+ (1 − 2w )σjM
2
2
[w 2 σj + (1 − w )2 σM + 2w (1 − w )σjM ]1/2
)
Deriving the CAPM
f 0 (σP ) = g 0 (w )/h0 (w )
g 0 (w ) = E (˜
rj ) − E (˜
rM )
2
w σj2 − (1 − w )σM
+ (1 − 2w )σjM
h (w ) = 2 2
2
[w σj + (1 − w )2 σM
+ 2w (1 − w )σjM ]1/2
0
imply
f 0 (σP ) = [E (˜
rj ) − E (˜
rM )]
2 2
2
[w σj + (1 − w )2 σM
+ 2w (1 − w )σjM ]1/2
×
2
w σj2 − (1 − w )σM
+ (1 − 2w )σjM
Deriving the CAPM
The red curve is tangent to the CML at M. Hence, f 0 (σP )
equals the slope of the CML when w=0.
Deriving the CAPM
When w = 0,
f 0 (σP ) = [E (˜
rj ) − E (˜
rM )]
2
[w 2 σj2 + (1 − w )2 σM
+ 2w (1 − w )σjM ]1/2
×
2
w σj2 − (1 − w )σM
+ (1 − 2w )σjM
implies
f 0 (σP ) =
[E (˜
rj ) − E (˜
rM )]σM
2
σjM − σM
Meanwhile, we know that the slope of the CML is
E (˜
rM ) − rf
σM
Deriving the CAPM
The tangency of the red curve with the CML at M therefore
requires
[E (˜
rj ) − E (˜
rM )]σM
E (˜
rM ) − rf
=
2
σjM − σM
σM
2
[E (˜
rM ) − rf ][σjM − σM
]
2
σM
σjM
E (˜
rj ) − E (˜
rM ) =
[E (˜
rM ) − rf ] − [E (˜
rM ) − rf ]
2
σM
σjM
E (˜
rj ) = rf +
[E (˜
rM ) − rf ]
2
σM
E (˜
rj ) − E (˜
rM ) =
Deriving the CAPM
E (˜
rj ) = rf +
σjM
2
σM
Let
βj =
[E (˜
rM ) − rf ]
σjM
2
σM
so that this key equation of the CAPM can be written as
E (˜
rj ) = rf + βj [E (˜
rM ) − rf ]
where βj , the “beta” for asset j, depends on the covariance
between the returns on asset j and the market portfolio.
Deriving the CAPM
E (˜
rj ) = rf + βj [E (˜
rM ) − rf ]
This equation summarizes a very strong restriction.
It implies that if we rank individual stocks or portfolios of
stocks according to their betas, their expected returns should
all lie along a single security market line with slope E (˜
rM ) − rf .
Deriving the CAPM
According to the CAPM, all assets and portfolios of assets lie
along a single security market line. Those with higher betas
have higher expected returns.
Deriving the CAPM
There are two complementary ways of interpreting this result.
Both bring us back to the theme of diversification emphasized
by MPT.
Both take us a step further, by emphasizing as well the idea of
aggregate risk, which cannot be “diversified away,” and
idiosyncratic risk, which can be diversified away.
Deriving the CAPM
The first approach uses the CAPM equation in its original form
σjM
E (˜
rj ) = rf +
[E (˜
rM ) − rf ]
2
σM
together with the definition of correlation, which implies
ρjM =
σjM
σj σM
to re-express the CAPM relationship as
E (˜
rM ) − rf
ρjM σj
E (˜
rj ) = rf +
σM
Deriving the CAPM
E (˜
rM ) − rf
E (˜
rj ) = rf +
ρjM σj
σM
The term inside brackets is the equilibrium price of risk.
And since the correlation lies between −1 and 1, the term
ρjM σj , satisfying
ρjM σj ≤ σj ,
represents the “portion” of the total risk σj in asset j that is
correlated with the market return.
Deriving the CAPM
E (˜
rM ) − rf
E (˜
rj ) = rf +
ρjM σj
σM
The idiosyncratic risk in asset j, that is, the portion that is
uncorrelated with the market return, can be diversified away by
holding the market portfolio.
Since this risk can be freely shed through diversification, it is
not “priced.”
Deriving the CAPM
E (˜
rM ) − rf
E (˜
rj ) = rf +
ρjM σj
σM
Hence, according to the CAPM, risk in asset j is priced only to
the extent that it takes the form of aggregate risk that,
because it is correlated with the market portfolio, cannot be
diversified away.
Deriving the CAPM
E (˜
rM ) − rf
ρjM σj
E (˜
rj ) = rf +
σM
Thus, according to the CAPM:
1. Only assets with random returns that are positively
correlated with the market return earn expected returns
above the risk free rate. They must, in order to induce
investors to take on more aggregate risk.
Deriving the CAPM
E (˜
rM ) − rf
E (˜
rj ) = rf +
ρjM σj
σM
Thus, according to the CAPM:
2. Assets with returns that are uncorrelated with the market
return have expected returns equal to the risk free rate,
since their risk can be completely diversified away.
Deriving the CAPM
E (˜
rM ) − rf
E (˜
rj ) = rf +
ρjM σj
σM
Thus, according to the CAPM:
3. Assets with negative betas – that is, with random returns
that are negatively correlated with the market return –
have expected returns below the risk free rate! For these
assets, E (˜
rj ) − rf < 0 is like an “insurance premium” that
investors will pay in order to insulate themselves from
aggregate risk.
Deriving the CAPM
The second approach to interpreting the CAPM uses
E (˜
rj ) = rf + βj [E (˜
rM ) − rf ]
together with the definition of
βj =
σjM
2
σM
Deriving the CAPM
Consider a statistical regression of the random return r˜j on
asset j on a constant and the market return r˜M :
r˜j = α + βj r˜M + εj
This regression breaks the variance of r˜j down into two
“orthogonal” (uncorrelated) components:
1. The component βj r˜M that is systematically related to
variation in the market return.
2. The component εj that is not.
Do you remember the formula for βj , the slope coefficient in a
linear regression?
Deriving the CAPM
Consider a statistical regression of the random return r˜j on
asset j on a constant and the market return r˜M :
r˜j = α + βj r˜M + εj
Do you remember the formula for βj , the slope coefficient in a
linear regression? It is
βj =
σjM
2
σM
the same “beta” as in the CAPM!
Deriving the CAPM
Consider a statistical regression:
2
r˜j = α + βj r˜M + εj with βj = σjM /σM
the same “beta” as in the CAPM!
But this is not an accident: to the contrary, it restates the
conclusion that, according to the CAPM, risk in an individual
asset is priced – and thereby reflected in a higher expected
return – only to the extent that it is correlated with the
market return.
Valuing Risky Cash Flows
We can also use the CAPM to value risky cash flows.
Let C̃t+1 denote a random payoff to be received at time t + 1
(“one period from now”) and let PtC denote its price at time t
(“today.”)
If C̃t+1 was known in advance, that is, if the payoff were
riskless, we could find its value by discounting it at the risk
free rate:
C̃t+1
PtC =
1 + rf
Valuing Risky Cash Flows
But when C̃t+1 is truly random, we need to find its expected
value E (C̃t+1 ) and then “penalize” it for its riskiness either by
discounting at a higher rate
PtC =
E (C̃t+1 )
1 + rf + ψ
or by reducing its value more directly
PtC =
E (C̃t+1 ) − Ψ
1 + rf
Valuing Risky Cash Flows
PtC =
E (C̃t+1 )
1 + rf + ψ
E (C̃t+1 ) − Ψ
1 + rf
The CAPM can help us identify the appropriate risk premium
ψ or Ψ.
PtC =
Our previous analysis suggests that, broadly speaking, the risk
premium implied by the CAPM will somehow depend on the
extent to which the random payoff C̃t+1 is correlated with the
return on the market portfolio.
Valuing Risky Cash Flows
To apply the CAPM to this valuation problem, we can start by
observing that with price PtC today and random payoff C̃t+1
one period from now, the return on this asset or investment
project is defined by
1 + r˜C =
or
r˜C =
C̃t+1
PtC
C̃t+1 − PtC
PtC
where the notation r˜C emphasizes that this return, like the
future cash flow itself, is risky.
Valuing Risky Cash Flows
Now the CAPM implies that the expected return E (˜
rC ) must
satisfy
E (˜
rC ) = rf + βC [E (˜
rM ) − rf ]
where the project’s beta depends on the covariance of its
return with the market return:
βC =
σCM
2
σM
This is what takes skill: with an existing asset, one can use
data on the past correlation between its return and the market
return to estimate beta. With a totally new project that is just
being planned, a combination of experience, creativity, and
hard work is often needed to choose the right value for βC .
Valuing Risky Cash Flows
But once a value for βC is determined, we can use
E (˜
rC ) = rf + βC [E (˜
rM ) − rf ]
together with the definition of the return itself
r˜C =
C̃t+1
−1
PtC
to write
E
!
C̃t+1
− 1 = rf + βC [E (˜
rM ) − rf ]
PtC
Valuing Risky Cash Flows
!
E
C̃t+1
−1
PtC
= rf + βC [E (˜
rM ) − rf ]
implies
1
PtC
E (C̃t+1 ) = 1 + rf + βC [E (˜
rM ) − rf ]
PtC =
E (C̃t+1 )
1 + rf + βC [E (˜
rM ) − rf ]
Valuing Risky Cash Flows
Hence, through
PtC =
E (C̃t+1 )
1 + rf + βC [E (˜
rM ) − rf ]
the CAPM implies a risk premium of
ψ = βC [E (˜
rM ) − rf ]
which, as expected, depends critically on the covariance
between the return on the risky project and the return on the
market portfolio.
Valuing Risky Cash Flows
Alternatively,
E (˜
rC ) = rf + βC [E (˜
rM ) − rf ]
and
r˜C =
C̃t+1
−1
PtC
imply
E
and hence
1
PtC
!
C̃t+1
− 1 = rf + βC [E (˜
rM ) − rf ]
PtC
E (C̃t+1 ) = 1 + rf + βC [E (˜
rM ) − rf ]
Valuing Risky Cash Flows
1
PtC
E (C̃t+1 ) = 1 + rf + βC [E (˜
rM ) − rf ]
can be rewritten as
PtC =
E (C̃t+1 ) − PtC βC [E (˜
rM ) − rf ]
1 + rf
indicating that the CAPM also implies
Ψ = PtC βC [E (˜
rM ) − rf ]
which, again as expected, depends critically on the covariance
between the return on the risky project and the return on the
market portfolio.
Strengths and Shortcomings of the CAPM
An enormous literature is devoted to empirically testing the
CAPM’s implications.
Although results are mixed, studies have shown that when
individual portfolios are ranked according to their betas,
expected returns tend to line up as suggested by the theory.
Strengths and Shortcomings of the CAPM
A famous article that presents results along these lines is by
Eugene Fama (Nobel Prize 2013) and James MacBeth, “Risk,
Return, and Equilibrium,” Journal of Political Economy Vol.81
(May-June 1973), pp.607-636.
Early work on the MPT, the CAPM, and econometric tests of
the efficient markets hypothesis and the CAPM is discussed
extensively in Eugene Fama’s 1976 textbook, Foundations of
Finance.
Strengths and Shortcomings of the CAPM
More recent evidence against the CAPM’s implications is
presented by Eugene Fama and Kenneth French, “Common
Risk Factors in the Returns on Stocks and Bonds,” Journal of
Financial Economics Vol.33 (February 1993): pp.3-56.
This paper shows that equity shares in small firms and in firms
with high book (accounting) to market value have expected
returns that differ strongly from what is predicted by the
CAPM alone.
Quite a bit of recent research has been directed towards
understanding the source of these “anomalies.”
Strengths and Shortcomings of the CAPM
Despite some empirical shortcomings, however, the CAPM
quite usefully deepens our understanding of the gains from
diversification.
Related, the CAPM alerts us to the important distinction
between idiosyncratic risk, which can be diversified away, and
aggregate risk, which cannot.
Strengths and Shortcomings of the CAPM
Like MPT, the CAPM must rely on one of the two strong
assumptions – either quadratic utility or normally-distributed
returns – that justify mean-variance utility.
And while the CAPM is an equilibrium theory of asset pricing,
it stops short of linking asset returns to underlying economic
fundamentals.
These last two points motivate our interest in other asset
pricing theories, which are less restrictive in their assumptions
and/or draw closer connections between asset prices and the
economy as a whole.
Strengths and Shortcomings of the CAPM
These last two points motivate our interest in other
asset-pricing theories, which are less restrictive in their
assumptions and/or draw closer connections between asset
prices and the economy as a whole.
Arbitrage Pricing Theory, to which we will turn our attention
next, yields many of the same implications as the CAPM, but
requires less restrictive assumptions about preferences and the
distribution of asset returns.
Strengths and Shortcomings of the CAPM
These last two points motivate our interest in other
asset-pricing theories, which are less restrictive in their
assumptions and/or draw closer connections between asset
prices and the economy as a whole.
The equilibrium version of Arrow-Debreu theory draws links
between asset prices and the economy that are only implicit in
the CAPM.